Model of Gas Flow Through Porous Refractory Applied to an Upper Tundish Nozzle

Abstract

Argon gas commonly is injected into the liquid metal stream through the porous refractory walls in many metallurgical processes. In this work, a new model has been developed to investigate gas diffusion through heated porous refractory, including the effects of refractory geometry, the thermal expansion of the gas, temperature-dependent gas viscosity, and possible leakage into unsealed joints. A novel one-way-flow pressure boundary condition has been formulated and implemented to prevent unrealistic flow into the refractory. The complete model is validated with both analytical solutions of 1D test problems and observations of a water bubbling experiment. Then, to demonstrate practical application of this general model, argon gas flow is simulated through a double-slitted upper tundish nozzle during continuous steel casting with a slide-gate system. Realistic liquid steel pressure distributions with the bubbling threshold condition are applied on the inner surface. Parametric studies are conducted to investigate the effects of joint gas leakage, refractory conductivity, permeability, and injection pressure on the resulting gas distributions, gas mass flow rates, and leakage fraction. This new model of porous flow can serve as the first step of a comprehensive multiphase model system.

Appendix A. Analytical Solution for 1D Test Problem

The one-way coupled heat conduction and pressure-source Eqs. [2] and [9] simplify into two ODEs, expressed in cylindrical coordinates as Eqs. [A1] and [A4]. The heat conduction equation,

$$ \frac{1}{r}\left( {rT^{\prime}} \right)^{\prime } = 0, $$

(A1)

has the general solution,

$$ T = C_{1} \ln r + C_{2} . $$

(A2)

The two integration constants, C ₁ and C ₂, are determined from the fixed temperature boundary conditions at the inner (T ₁) and outer radius (T ₂):

$$ C_{1} = \frac{{T_{2} - T_{1} }}{{\ln \left( {R_{2} /R_{1} } \right)}}\quad {\text{and}}\quad C_{2} = \frac{{T_{1} \ln R_{2} - T_{2} \ln R_{1} }}{{\ln \left( {R_{2} /R_{1} } \right)}}. $$

(A3)

The 1D gas pressure equation can be expressed as:

$$ p^{\prime\prime} + \left( {\frac{1}{r} - \frac{{T^{\prime}}}{T} + \frac{{K^{\prime}_{\text{D}} }}{{K_{\text{D}} }}} \right)p^{\prime} + \frac{{p^{'2} }}{p} = 0. $$

(A4)

Equation [A4] was discretized using a central finite-difference scheme and solved with a tri-diagonal matrix algorithm (TDMA), on a 200-node mesh.[42] Equation [A4] can be solved analytically for two special cases: (1) with gas thermal expansion but constant gas viscosity; and (2) without any thermal effects.

With Thermal Expansion and Constant Gas Viscosity

With constant gas viscosity, the Equation [A4] simplifies to:

$$ pp^{\prime\prime} + p^{'2} + pp^{\prime}\left( {\frac{1}{r} - \frac{{T^{\prime}}}{T}} \right) = 0 $$

(A5)

which has the following positive general solution:

$$ p = \sqrt {\frac{{C_{3} }}{{C_{1} }}\left( {C_{1} \ln r + C_{2} } \right)^{2} + 2C_{4} } . $$

(A6)

Applying Eq. [8] gives the corresponding gas velocity:

$$ V_{\text{r}} = - \frac{{K_{\text{D}} \sqrt {C_{1} C_{3} } \left( {C_{1} \ln r + C_{2} } \right)}}{{r\sqrt {\left( {C_{1} \ln r + C_{2} } \right)^{2} + \frac{{2C_{1} C_{4} }}{{C_{3} }}} }}, $$

(A7)

where integration constants C ₁ and C ₂ are given in Eq. [A3]. Inserting the fixed pressure boundary conditions, at the inner (P ₁) and outer radius (P ₂) gives:

$$ C_{3} = \frac{{P_{2}^{2} - P_{1}^{2} }}{{T_{2}^{2} - T_{1}^{2} }}C_{1} \;{\text{and}}\;C_{4} = \frac{{P_{1}^{2} T_{2}^{2} - P_{2}^{2} T_{1}^{2} }}{{2\left( {T_{2}^{2} - T_{1}^{2} } \right)}}. $$

(A8)

No Thermal Effects

The simplest scenario ignores both temperature-dependent gas viscosity and gas expansion. Assuming constant viscosity and temperature simplifies Eq. [A4] into:

$$ p^{\prime\prime} + \frac{{p^{\prime}}}{r} = 0 $$

(A9)

which has the pressure solution:

$$ p = C_{3} \ln r + C_{4} , $$

(A10)

where,

$$ C_{3} = \frac{{P_{2} - P_{1} }}{{\ln \left( {R_{2} /R_{1} } \right)}}\;{\text{and}}\;C_{4} = \frac{{P_{1} \ln R_{2} - P_{2} \ln R_{1} }}{{\ln \left( {R_{2} /R_{1} } \right)}}. $$

(A11)

The corresponding gas velocity distribution is: $V_{\text{r}} = - K_{\text{D}} \frac{{C_{3} }}{r}$.